Question

Question 4: Talyor. Maclaurin and Power Series For parts a, b, c and d, use the following function: f(x) = (-3x a) (3 points) Write the Taylor polynomial of degree four for f(x) centered at 0. b) (2 points) Use the Taylor polynomial from part a to estimate the value of e-0.3. (Hint: let find x). c) (3 points) Write the series generated by f(x) at zero in sigma notation. d) (3 points) Find the radius of convergence and state the interval of convergence. e) (4 points) Use the Maclaurin series to show that the following limit is e-3x – 1 + 3x lim k0 3x2

Answer 1

32 fen) -33 f(x) = e-3k f(o)-1 f'(x) = -3 e 3 = f'(o)= -3 f"(u) = 32 e 3 a f"o) = -33 e 30 = f'" cola flv caja 34 e 3 - Scola 34 frol & scola fitos only ㅠ -31 +3² 24 f(n) = + 1 W3 + uy 22 - 3 3. is founth degree daylon semes 6 e-0.3 3(-0.19 fail) 1-310.10 + 3² ) (0.13 (0.12 + + 3 पा CODY F1 -0.3+ (0.01) -23 (0-60) 24 (0.6) = 1-0.3 +0.045 + 81 0.0045 + 0.0003375 0.7408375 flu) - § (-17%. 3 x no D here an= (n. 30 anti (int 3n41 reding of Convengene um Coty cim (123011). anll un n10 2020

20207773 3 x t- ttak hence reding of Convengence is as zintenval of Convengence = (-00100) 1-31 um e -1 +32 Kto 3 u2 + Kho 34€ = 3/1 = 6/2

I use definition of Taylor series and maclerian series to solve this problem